I've been wondering what the period can be of a planet or moon oriting at the hill sphere at distance r . http://orbitsimulator.com/gravity/articles/hillsphere.html gives the radius (or Sma) . Amazingly the period t of the orbiting moon is easily derived as : t= T * sqrt ( (1-e)^3)/3). or in case of a circular orbit of the planet as : t= T * sqrt (1/3) . In case of the Sun-Earth-Moon system the t becomes : 1 year/1.73 or about 7 months!.

it has been know that the maximum extent of stability was between 0.33 and 0.5 x Hill Sphere

Using the Gas Giants on can see this range. If you simulate the the Jovian system in it's entirety you will see that over geologic time the outermost retrograde moon is actually in an unstable orbit it is just beyond the retrograde stability zone

Retrograde moons with axes up to 67% of Hill radius are believed to be stable.

This statement from the article is disputable as there is not reference to it give in the article. and there is another article that better explains the Hill Sphere http://en.wikipedia.org/wiki/Hill_sphere#True_region_of_stability Using simulations of n bodies, HillSphere * (Phi/e)^Phi was the maximum range That I could keep a retrograde orbit stable indefinitely.

I have a question about the phrase in the link above : "Once the hill sphere has been determined next we need to determine the semi-major axis outer boundary for pro-grade orbits; this is done by taking the hill sphere * ( Phi / e ) ^ Phi ^ ( e / 2 ). Just to insure we have all necessary data points we should also calculate the semi-major axis outer boundary for retrograde orbits; this is done by taking the hill sphere * ( Phi / e ) ^ Phi. " I have known the Hill sphere as an upper limit of stable bound orbits , but I'm curious to see how te above formulae were derived ?

I think the article in Wiipedia mentioning a maximum stabilty region to be between 1/2 and 1/3 of the Hillradius is acceptable . I've checked wit GravitySimulator the stability for the Sun-Jupiter system doing the following : Created the Sun-Jupiter system , added 300 zero masses at 0.15AU+/-50% around Jupiter and let run for ca. 40 years , at a small time-step. A lot of exit the system , but about 20% stay stable. The result of the output in Excel ( after deleting the instable ones ) shows the variation of the SMA of the bodies which can be accepted as being "stable". The more closer they start the less variation in SMA occurs. Looking at the picture I guess one can attribute a maximum SMA of about 22.000.000 km for this system , or perhaps a little bit more . This corresponds with 0,41 rHill.

BTW : running this system makes really fun. One can see the ring of bodies becoming elleptical , keeping a circular "bulge" . Instable ones leave the system creating "arms" . Looks like a small galaxy. If wanted i can provide the .gsim .

I derived them empirically by plotting out the data and performing curve fitting the best curve fit I could achieve where those that are the result listed in the paper. It was not through any advanced mathematical process.

the outer boundry for prograde results in a constant value close to 0.319535834655967 the outer boundry for retrograde results in a constant value close to 0.431962327235041

Here's an animation of a gsim simulation about the Hill Sphere . Around Jupiter (in the center) are orbiting 300 1kg masses in retrograde orbit . Orbits are initially circular , at random positionned at 0.2 AU +0.2 . One sees the system quickly evolve towards an elliptical system and even to a system with spiral arms at opposite sides of the central Jupiter . At first I wondered why the spiral arms are created at both sides , but then I realised this is due to the "tidal" effects of the sun. This is at first glance contra-intuitive . I have the impression that a retrograde system allows much more eccentricity in the orbits than a prograde system .

BTW ; is there any way to attach also the .gsim file in the same post ? The system doesn't seem to accept multiple attachments.